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Symbolic solution to expansion of natural frequencies method for determining

Written by Nasser M. Abbasi, based on lecture given by Professor Barmish, ECE 717, University of Wisconsin, Madison

11/1/2014

The method of finding using expansion of natural frequencies is first applied to a *2**×**2* matrix to illustrate the symbolic method. Then a general purpose function is written at the end using the symbolic method and applied to much larger problems for illustrations. This is a standalone function which can be called to determine for matrices A which can have repeated eigenvalues. The result is verified by comparing the output to Mathematica buildin function MatrixExp[]

Example 1 from lecture done using symbolics

Define *A*

Find eigenvalues of A

Set the eigenvalue multiplers

Apply algorithm using expansion of natural frequencies

Automatic generation of lists of unknowns Y(k,i) and list of symbols to b[i] to use for solving Ax=b

Take repeated derivatives of to generate the set of equations to solve for Y(k,i)

The problem can be viewed as Ax=b where x=Y(k,i) is the unknowns and b are the symbolic names for the matrix powers.

Now solve Ax=b

Replace symbols b(i) on the right of the above, by the actual numerical values of the matrix powers

Now apply these numerical values to the solution found earlier in order to obtain numerical values for Y(k,i)

Now since Y(k,i) are now found, evaluate found above using the above solution

Compare to Mathematica buildin function

Function to automatically calculate using symbolic natural frequencies method

Example 2 from lecture solved using the above function

λ | multiplier |

3 | 1 |

2 | 3 |

1 | 2 |

Compare to buildin function to verify

Verification Method, suggested by Professor Barmish. We Differentiate Matrix found above and see it is the same as for *t**=**0*

k | ||

0 | ||

1 | ||

2 | ||

3 | ||

4 | ||

5 |